On the dual of complex Ol'shanskiĭ semigroups (Q5947797)

The author investigates the relationships between three natural topologies \({\mathcal T}^\alpha_{hk}\), \({\mathcal T}^\alpha_G\), and \({\mathcal T}^\alpha_e\) on the set \(\widehat S_\alpha\) of equivalent classes of irreducible \(\alpha\)-bounded representations of a complex Ol'shanskiĭ semigroup \(S\), where \(\alpha\) is an absolute value on \(S\), i.e., a map \(\alpha:S\to[0,\infty)\) such that \(\alpha(\bar s^{-1})=\alpha(s)\) and \(\alpha(st)\leq\alpha(s)\alpha(t)\) for any \(s,t\in S\). A complex Ol'shanskiĭ semigroup is a semigroup of the form \(S=G \exp(iW)\subset G_{\mathbb G}\) in the universal complexification \(G_{\mathbb G}\) of a connected Lie group \(G\) for some non-empty open \(Ad(G)\)-invariant convex cone \(W\) in the Lie algebra of \(G\), containing no affine lines. The main result asserts that \({\mathcal T}^\alpha_{hk}\subset{\mathcal T}^\alpha_G\subset{\mathcal T}^\alpha_e\). These three topologies induce the same Borel structure on \(\widehat S_\alpha\) and for generic absolute values they coincide. Here \({\mathcal T}^\alpha_{hk}\) is the hull-kernel topology induced by identification of \(\widehat S_\alpha\) with the set of non-degenerate representations of a certain \(C^*\)-algebra \(C^*_h(S,\alpha)\); the topology \({\mathcal T}^\alpha_G\) is induced by the topology of the dual group \(\widehat G\) under an identification of \(\widehat S_\alpha\) with a subset of \(\widehat G\). Finally, the topology \({\mathcal T}^\alpha_e\) is induced by the Euclidean topology of a certain subset of highest weights \(HW_\alpha\), parametrizing the set \(\widehat S_\alpha\).